APPENDIX: COPULA FUNCTION AND COPULA DENSITY
Deriving the Copula Density from Copula Function
Since the copula function C is a joint CDF taking as arguments the CDF functions derivatives with respect to x and y, implies using the chain rule for derivatives: Taking first the derivative with respect to arguments of C, F and F2, which are functions of x and y, and multiplying each by the derivatives of F (x), F (y) with respect to their arguments x and y.
The chain rule for derivatives using two functions g and h of x, such that k(x) — g[h(x)] implies that:
In the joint CDF, the copula function C is the equivalent of g and the joint CDF F(x) is the equivalent of h(x). Hence with a single function of x, we would write:
The rule applies as well to two functions:
It should be kept in mind that the CDFs F are themselves functions of x and y, so that the expanded expression of C is:
Joint Probability Density Functions with Two Variables
The copula function is a joint CDF, equal to the probabilities that the two variables are lower than or equal to x and y, respectively. The joint probability density function, or joint PDF, is the probability that two variables following the distributions functions F (x) and F2(x2) take the joint pair of values x and y. probability (X—x and Y — y) — joint PDF(x, y)
Starting from the copula function:
Because it cumulates the probabilities of Xand F being lower than x and y, it sums up all joint probabilities of couples of values of X and Flower than x and y. The copula, being a cumulative function, is the integral of the corresponding joint density function that applies to a couple of values x and y. This joint PDF, or joint probability applicable to the pair (x, y), is derived from the copula function by taking the derivatives with respect to the two values x and y, or the second derivatives with respect to x and y respectively:
The derivation of this expression is expanded in the next section.
When the CDF of x is F(x), the PDF of x is fix) — Fx). Moreover, for two variables, the joint PDF is simply fix, y). This makes the notations a bit easier to follow. The two univariate density functions are/^(x) = F^x) and f2(y) = F2(y). Putting those notations together, we get the joint density function as the product of the two density functions multiplied by the second derivatives of the copula C with respect to the CDFs F and F2:
It is convenient to use as abbreviation for this second derivative of the copula C, the small letter c, defined as:
c(Fv F2) is the copula density. The copula density provides the joint density function of variables Xand Fas the copula density multiplied by the marginal (unconditional and univariate) densities (or probabilities) of X and Y, F (x), F (y). The copula density is a function of the joint density of two variables and their unconditional (marginal) densities.
The copula density is:
This provides the intuitive understanding of the copula density, as the ratio of the joint probability density to the product of the probabilities that X— x and Y — y. The product is equal to the joint probability of x and y occurring only when Xand Y are independent.
• When X and Fare independent, the copula density takes the value 1.
• When X and Fare positively correlated, the copula density is higher than 1 because the joint probability fix,y) >fl(x)f2(y).
• When X and Fare inversely related, the copula density is lower than 1 because fix, y) < fx(x)